Lattice QCD for Cold Nuclear Physics - USQCD: US Lattice ... · Lattice QCD for Cold Nuclear...

Lattice QCD for Cold Nuclear Physics

John Negele, David Richards and Martin J. Savage

(USQCD Collaboration)

(Dated: February 22, 2013)

CONTENTS

I. EXECUTIVE SUMMARY

II. INTRODUCTION

III. COMPUTATIONAL RESOURCES FOR LATTICE QCD

IV. HADRON STRUCTURE

IVA. Motivation and Achievements

IVB. Future Program and Resource Requirements

V. HADRON SPECTROSCOPY

VA. Motivation and Achievements

VB. Future Program and Resource Requirements

VI. HADRONIC INTERACTIONS, NUCLEAR FORCES AND NUCLEI

VIA. Motivation and Achievements

VIB. Future Program and Resource Requirements

VII. FUNDAMENTAL SYMMETRIES

VIIA. Motivation and Achievements

VIIB. Future Program and Resource Requirements

VIII. SUMMARY

I. EXECUTIVE SUMMARY

Quantum chromodynamics (QCD) is the theory of the strong interactions which, alongwith the electroweak interactions, underpins all of nuclear physics and the vast array ofphenomena, from the microscopic to the cosmological, that nuclear physics impacts. Thisdocument summarizes the current status of the application of Lattice QCD to the calculationof quantities that are of central importance to science, including the structure of the nucleon,the spectrum of the mesons and baryons, the interactions between nucleons, the spectrumand structure of the light-nuclei, and probes of fundamental symmetries in nuclear physics.The planned programs, along with the anticipated computational resource requirements, areoutlined in each of these areas, and the connections between the nuclear physics experimentalprogram and the Nuclear Science Advisory Committee (NSAC) Performance Measures[1] arehighlighted.

It is anticipated that the next five years of research in this area will include the first (Lattice)QCD calculations of low-energy nuclear physics quantities of central importance to thefield at the physical light-quark masses and with fully quantified uncertainties, therebyfundamentally changing the field of nuclear physics. This transformative change to the fieldis predicated on Moore’s Law growth in available computational resources. Indeed, suchhas been the remarkable progress made by USQCD during the preceding five years that thisnew and emerging methodology could exploit resources growing faster than Moore’s law toaccelerate still further our understanding of nuclear physics.

II. INTRODUCTION

The strong interaction is responsible for a diverse range of physical phenomena, includingbinding gluons and the lightest quarks into pions, protons, and neutrons. Through itsmanifestation as the strong nuclear force, it then binds neutrons and protons together toform the elements of the periodic table. The strong nuclear force plays a central role in thefusion of light nuclei, which occurs within the sun’s core. Within certain heavy nuclei, thestrong nuclear force, in a delicate balance with the electromagnetic interaction, is responsiblefor the process of fission that is exploited, for instance, in nuclear power stations.

A key issue is the phenomenon of confinement that is a well-established outcome of QCD,but the exact nature of confinement and all of its consequences are still a mystery. Thus,how does QCD give us the particular mesons and baryons we observe, and how are gluonsmanifest in the spectrum? In nucleons, how do quarks and gluons interacting through QCDproduce their observed mass, size, and spin? Similar questions remain unanswered about theexact nature of the strong nuclear force and its lineage from QCD. For example, what is thenature and origin of the nuclear spin-orbit interaction? How does the three-neutron forceemerge from QCD? Even partial answers to these questions will help illuminate long-standingissues in nuclear physics and ultimately provide a deeper understanding of how protons andneutrons combine to form nuclei, and therefore the elements germane to everyday life.

The breadth of these open issues does not represent a lack of progress in understandingnature in these environments. Rather, computational resources are now reaching a point

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where answers to many of these questions are within the realm of possibility over the nextfive years. Already, substantial progress has been made in using available computational re-sources to investigate these questions. As computational research in these areas matures, weexpect still further progress in exploiting the increasingly powerful computational resourcesthat will be available.

Accomplishments over the last five years have established the methodology and the ground-work for lattice investigations of hadron structure, spectroscopy, and the structure of thelightest nuclei and their interactions. Calculations have been performed of the strong-interaction effects needed to unveil the fundamental symmetries of the Standard Modelfrom precision nuclear-physics experiments. An impressive range of calculations has beenperformed at decreasingly smaller values of the light-quark masses. However, present daycomputational-resource limitations restrict the mass of the pion input into those calculationsto one exceeding that of nature.

The next five years will be a transformational period for nuclear physics. The 12 GeVupgrade of Jefferson Laboratory will explore the role of gluonic excitations in the QCDspectrum, provide a three-dimensional tomographic view of the proton, and through parity-violating electron scattering explore the fundamental symmetries of the Standard Model.Experiments at RHIC-spin will probe the gluon and antiquark contributions to the spin of theproton, and experiments at JLab will explore the contribution of orbital angular momentum.The construction of the Facility for Rare Isotope Beams will enable the exploration of multi-nucleon interactions from neutron-rich nuclei. Provided sufficient computational resourcesare made available, precision first-principles calculations of the spectrum, structure andinteractions of the building blocks of all the visible matter in our universe will be performedwith fully quantified uncertainties.

The centrality of computational physics in providing new understandings of, and predictivecapabilities for, nuclear physics was recognised in a recommendation of the NRC decadalreview of nuclear physics Nuclear Physics: Exploring the Heart of Matter [2]:

A plan should be developed within the theoretical community and enabledby the appropriate sponsors that permits forefront computing resources to bedeployed by nuclear science researchers and establishes the infrastructure andcollaborations needed to take advantage of exascale capabilities as they becomeavailable.

Not only will this provide the first reliable connection between QCD and low-energy nuclearphysics, but it will also provide the tools to explore and ultimately understand the fine-tunings that, in some sense, define matter.

In the remainder of this paper, we begin by outlining the computational resources availableto the USQCD collaboration, the essential role of software development facilitated throughSciDAC, and our assumptions about growth in computational resources over the next fiveyears. We will then describe the opportunities for Lattice QCD to advance our understandingacross four key campaigns in hadronic and nuclear physics before concluding with a summary.

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III. COMPUTATIONAL RESOURCES FOR LATTICE QCD

At present, members of USQCD are making use of dedicated hardware funded by the DOEthrough the LQCD-ext Computing Project, as well as a Cray XE/XK computer, and IBMBlue Gene/Q and Blue Gene/P computers, made available by the DOE’s INCITE Program.During 2013, USQCD, as a whole, expects to attain approximately 300 Tflops-years onthese machines. USQCD has a PRAC grant for the development of code for the NSF’spetascale computing facility, Blue Waters, and expects to obtain a significant allocation onthis computer during 2013. Subgroups within USQCD also have allocations at the DOE’sNational Energy Research Scientific Computing Center (NERSC) and through the NSF’sXSEDE Program, access to dedicated Blue Gene/Q computers at Brookhaven NationalLaboratory and the University of Edinburgh, access to the Blue Gene/Q computers atForschungszentrum JUlich, and access to computers at LLNL.

For some time, the resources we have obtained have grown with a doubling time of approx-imately 1.5 years, consistent with Moore’s law, and this growth rate will need to continueif we are to meet our scientific objectives. The software developed by USQCD under ourSciDAC grant enables us to use a wide variety of architectures with very high efficiency,and it is critical that our software efforts continue at their current pace. Notably, many ofthe calculations we outline below were enabled through our ability to exploit GPUs on theUSQCD clusters for the most demanding parts of the calculations. Over time, the develop-ment of new algorithms has had at least as important an impact on our field as advances inhardware, and we expect this trend to continue, although the rate of algorithmic advancesis not as smooth or easy to predict as that of hardware.

IV. HADRON STRUCTURE

A. Motivations and Achievements

One of the great challenges posed by QCD is understanding how protons and neutrons, thebasic building blocks of most of the observed matter in the universe, are made from quarksand glue. Just as calculating the structure of atoms was a cornerstone of quantum mechan-ics, a cornerstone of contemporary nuclear physics is to achieve a quantitative, predictiveunderstanding of the structure of nucleons and other hadrons using lattice QCD.

One goal in hadron structure is precision calculation of fundamental quantities characteriz-ing the nucleon, including form factors, moments of parton density, helicity, and transversitydistributions, moments of generalized parton distributions (GPDs) and transverse momen-tum distributions. Hadronic observables calculated from first principles are directly relevantto the experimental programs at JLab and RHIC-spin, for experimental data from SLAC andFNAL, and will have significant impact on future experiments at the JLab 12 GeV upgradeand at a planned electron-ion collider. Another essential goal is obtaining insight into howQCD works. How does the spin of the nucleon arise from the helicity and orbital angularmomentum of quarks and of gluons? What is the fundamental mechanism for confinementand what is the role of diquarks and instantons in hadrons? How does hadron structure

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change as one varies parameters that cannot be varied experimentally, such as the numberof colors, the number of flavors, the quark masses, or the gauge group? Finally, once latticecalculations are validated by extensive comparison with directly comparable experiments,one can exploit lattice QCD to make predictions relevant to planned experiments, to theresolution of experimental controversies, and to other areas of physics as discussed below.

Recent work has laid the groundwork for a broad range of important hadron structure cal-culations, subject to limitations in computational resources. Calculations have focused onthe dominant connected diagram contributions to matrix elements since the sub-dominantdisconnected contributions are much more computationally expensive. Fortunately, discon-nected contributions vanish for isovector quantities. Since the cost of lattice QCD calcula-tions in a volume large enough to contain a pion grows rapidly with the inverse pion mass,past calculations have been performed for a series of pion masses that have gradually ap-proached the physical mass and then been extrapolated to the physical mass using chiralperturbation theory. Due to uncertainty in the convergence of chiral perturbation theory,extrapolations from pion masses 300 MeV and above give rise to presently unquantifiedsystematic uncertainties. Also, calculations for low pion masses are susceptible to contami-nation by excited states that can cause apparent disagreement with experiment [3]. A recentcalculation including the nearly physical pion mass and using a method to reduce the contri-bution of excited states yielded agreement with experiment within statistical uncertaintiesfor several key isovector observables [4].

Electromagnetic form factors reveal the distribution of charge and current and how con-stituents interact to recoil together at high momentum transfer. Calculations[4–8] haveshown the emergence of the pion cloud at the periphery of the nucleon as the pion mass islowered. Calculations [4] at a pion mass of 149 MeV yield agreement with experiment forthe isovector charge radius, magnetization radius, and magnetic moment. The statisticalerror for the charge radius is comparable to the discrepancy between electron scattering andLamb shift measurements, so future high precision calculations will be important in helpingresolve this discrepancy.

The axial charge, gA, governing neutron β-decay, is the value of the axial form factor at zeromomentum transfer. Chiral extrapolations of calculations of gA at heavy quark masses agreewith experiment[5, 9, 10] with statistical uncertainties as low as 7%. However, calculationscloser to the physical pion mass with presently affordable spatial volumes, temporal extents,and excited state suppression are inconsistent with experiment[3, 4, 9]. Precision calculationsat a fraction of a percent will impact the proton-proton fusion rate central to solar modelsand constrain the weak matrix element |Vud|. The scalar and tensor charges, gS and gT ,corresponding to the scalar and tensor form factors at zero momentum transfer, are beingcalculated[11, 12] and are essential in searching for physics beyond the standard model inultra-cold neutron Beta-decay experiments at LANL. A precise prediction of the tensorcharge will be particularly significant prior to a major experimental program to measureit at the JLab 12 GeV upgrade. The precise knowledge of the strangeness content of thenucleon through the calculation of the strange quark scalar matrix element has implicationsfor the interactions of certain dark matter candidates with ordinary matter[13].

Quark parton distributions have been studied in deep-inelastic scattering experimentsaround the world and yield detailed data specifying quark density, spin, and transversitydistributions as functions of the momentum fraction, x, of the struck quark. The moments of

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1

2DS

u

1

2DS

d

Ld

Lu

0.0 0.1 0.2 0.3 0.4 0.5 0.6

-0.2

0.0

0.2

0.4

0.6

mΠ2@GeV

2D

co

ntr

ibu

tio

ns

ton

ucle

on

sp

in

FIG. 1. Contributions of quark spin ∆Σu,d/2 (blue error bars) and orbital angular momentum

Lu,d (red error bars) to the spin 1/2 of the proton for up and down quarks [5]. Chiral extrapolations

(shaded error bands) agree with HERMES data (crosses).

these distributions can be calculated with lattice QCD and to date, the dominant connecteddiagram contributions to the lowest three moments of each have been calculated[5, 14–16].When connected and disconnected contributions have been calculated to high precision, thecombination of a finite number of moments and experimental measurements over part ofthe range of x will determine these distributions better than either lattice calculations orexperiment can alone.

The so-called nucleon spin crisis refers to the fact that whereas in the naive quark model,all of the spin of the nucleon comes from the spins of the three quarks, experimentally onlya small fraction arises from quark spin. This issue is addressed by the lowest moment ofthe spin distribution, ∆Σ, which specifies the fraction of the nucleon’s spin arising from thespin of the quarks. As shown in Figure 1, chiral extrapolations of the calculations in Ref. [5]agree with deep inelastic scattering results from HERMES[17] and show that only 30% ofthe nucleon spin comes from quark spin. Thus, we see directly from solving QCD throughlattice calculations that the simple quark model is indeed quite naive.

The first moment of the quark density distribution, 〈x〉, specifies the fraction of the nucleonmomentum carried by quarks, and is well measured in deep inelastic scattering. The latticeQCD calculations of the isovector combination 〈x〉u−d have no disconnected contributionsand thus, when compared with experiment, provide an unambiguous test of current latticecalculations. Figure 2 shows a chiral fit to a lattice calculation[4] down to a pion mass of 149MeV that is in excellent agreement with the CTEQ6 analysis of deep inelastic scattering[98].Here one observes that the 149 MeV point highly constrains the fit so that extrapolation

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0.10 0.15 0.20 0.25mπ [GeV]

0.00

0.05

0.10

0.15

0.20

0.25〈x〉 u−

d

BChPT32c48 coarse48c48 coarseCTEQ6

FIG. 2. The isovector quark momentum fraction 〈x〉u−d [4] and the chiral fit (shaded error band)

agrees with the CTEQ6 result[98].

uncertainties are negligible.

Generalized Parton Distributions (GPD’s) specify quark density, spin, and transversity asfunctions of both the longitudinal momentum fraction x and the transverse position, and ifknown completely would enable us to calculate a 3-dimensional picture of the nucleon. Theirmoments in x have been calculated on the lattice[5, 19] and are in qualitative agreementwith phenomenology[20]. The combination of these moments with convolutions of GPDsmeasured at JLab and elsewhere will provide a more complete understanding than eithereffort could obtain separately.

Of particular interest and relevance to the DOE nuclear physics experimental program is thefact that one combination of moments of GPD’s specifies the total angular momentum ofquarks in the nucleon[5, 14, 18, 19, 21], so combined with the quark spin contribution above,the lattice provides a complete determination of quark spin and orbital contributions to thetotal spin 1/2 of the nucleon. The results [5] from the dominant connected diagrams shownin Figure 1 are striking. In addition to the fact mentioned above that only about 30% comesfrom quark spin, the substantial orbital contributions of up and down quarks cancel yieldinga negligible orbital contribution, so that the remaining 70% must come from sea quarksand gluons. This is of great interest experimentally and a major focus of RHIC-spin and afuture electron-ion collider is measuring the contribution of gluons to the nucleon spin. Directcalculation of the gluon contribution to the nucleon spin is clearly desirable. An important

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initial step was taken by calculating the contribution of gluons to the momentum fraction〈x〉 of the pion[22] and obtaining a result which when added to the quark contribution yields100% within uncertainties. A calculation of the gluon contribution to nucleon momentumand angular momentum has been made using a gluon energy-momentum tensor operatorderived from the overlap operator, in the expectation of reduced ultraviolet fluctuations[23].

Transverse momentum distributions show the quark momentum distribution, and initial cal-culations in a nucleon[24, 25] have opened the way for future results relevant to semi-inclusivedeep inelastic and single-spin asymmetry experiments at COMPASS, Hermes, JLab, and afuture electron-ion collider. Also, the spin-isospin excitation of the nucleon, the ∆(1232),has been shown to be deformed, consistent with phenomenology, by calculating its quarktransverse charge densities[26] and the transition form factors to it from the nucleon[27].

B. Future Program and Resource Requirements

Current computer resources and algorithms coupled with their expected improvements overthe next five years provide exciting opportunities for definitive precision computation ofpresently calculable observables as well as exploration of new observables and theoreticalissues.

• Precision computation of presently calculable observables

During the next five years, a major focus of the program will be precision computationof form factors, moments of quark parton distributions, and moments of quark GPDsat the physical pion mass with control of systematic uncertainties associated with finitelattice spacing, spatial volume, temporal extent and excited state contaminants. Thesecalculations are explicitly required to meet the 2014 NSAC Performance MeasureHP9, which states “Perform lattice calculations in full QCD of nucleon form factors,low moments of nucleon structure functions and low moments of generalized partondistributions including flavor and spin dependence”, and to address experimental mea-surements mandated by six of the other nine Performance Measures. These latticecalculations are essential for realizing the full physics potential of experiments at theJLab 12 GeV upgrade, RHIC-spin, and a planned electron-ion collider.

The methodology for calculating connected diagrams and removing excited state con-taminants is now sufficiently well-developed to enable estimation of the computationalcost for calculating these observables at the physical pion mass. Because of the ex-ponentially small signal-to-noise ratio at large times for measurements of nucleon ob-servables at the physical pion mass, the computational cost of the required number ofmeasurements on a configuration is much larger than the cost of generating the config-uration itself. Useful estimates can be obtained by taking present results very close tothe physical pion mass, scaling the cost to the physical pion mass and appropriate spa-tial volume, time extent, and parameters to reduce excited state contamination, andincreasing the number of measurements to reduce the statistical error to the desiredprecision. Cost estimates are given in Table I.

Disconnected diagrams, which describe processes in which the operator being calcu-lated excites a quark-antiquark pair that in turn interacts with one of the quarks in

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the nucleon, are computationally much more difficult and expensive to calculate thanconnected diagrams, but they are essential for precision calculations of the propertiesof protons and neutrons separately, and thereby to resolving the contributions of theindividual quark flavors to hadron structure. Although they have been calculated fora few operators, where their contributions are found to be smaller than those of thecorresponding connected contributions, they have never been calculated for the fullsuite of form factors, moments of quark parton distributions, and moments of quarkGPDs. Hence, it will be necessary to study the effectiveness of alternative dilutiontechniques and truncation methods to find the most efficient set of algorithms for afull calculation. In addition, it is necessary to compute the mixing of gluons withquark disconnected diagrams in order to calculate flavor singlet matrix elements. Thisentails a significant effort in developing an optimal gluon operator and calculating thenecessary renormalization factors and mixing coefficients. The ultimate computationalcost of calculating disconnected diagrams and the associated gluon mixing is presentlyunknown, but is expected to be more than an order of magnitude greater than that ofconnected diagrams.

• New Observables and Theoretical Issues

Much of the intellectual vitality and excitement of the field centers on new theoreticaldevelopments and ideas for calculating important physical quantities that are presentlyinaccessible. Some of the key problems that are important for further understandingthe quark-gluon structure of the nucleon are described below.

Form factors at high momentum transfer: At sufficiently high momentum transferq, asymptotic scaling sets in and the electric form factor falls off as q−4 and otherform factors fall off with other known powers. To determine this scale for asymptoticscaling, form factor calculations must be extended to high momentum transfer wherenew techniques[28] need to be explored and applied in large volumes at the physicalpion mass.

Higher moments of quark structure functions: Because the hypercubic group of thelattice has less symmetry than the continuum, unphysical operator mixing can occurin lattice QCD and current lattice QCD calculations only eliminate this mixing forthe lowest three moments of these distributions. To reconstruct quark distributions asfully as possible and to constrain phenomenological models, it is necessary to explorenew algorithms for calculating higher moments, such as introducing a heavy quarkfield [29], and using extended operators at sufficiently fine lattice spacings[30], and toexploit them at the physical pion mass.

Gluon contributions: Our understanding of nucleon structure will remain fundamen-tally incomplete until we learn to directly calculate gluon contributions to the nucleonmass, momentum, and angular momentum. Given the focus on measuring the gluoncontribution to the nucleon spin at RHIC-spin and a future electron-ion collider, thiswill have great impact on the experimental program. Since the fluctuations in gluonfields grow strongly as the lattice spacing is decreased, extracting the continuum limitfor gluon observables will require a combination of theoretical ideas, new algorithms,and techniques for obtaining extremely high statistics.

Transverse momentum distributions: It is important to relate lattice calculations oftransverse momentum distributions quantitatively to experimental measurements of

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semi-inclusive deep inelastic scattering and Drell-Yan processes. This requires access-ing the regime in which QCD factorization applies, which in turn requires developingtechniques to treat high nucleon momentum. Because of the rapid decrease of thesignal-to-noise ratio with momentum, extremely high statistics will be required forthese calculations as well.

TABLE I. Resources, in Tflops-years (TF-yrs), required to calculate the connected diagrams for a

complete set of form factors and the lowest three moments of structure functions and generalized

parton distributions for the nucleon at the physical pion mass. Lattice Generation is the cost of 104

trajectories, Measurements A reduce the error for gA to 3 % , and Measurements B reduce the error

for 〈x〉 to 3 %. The actions used are Wilson-clover fermions (W) and Domain-wall fermions (DW);

measurements for both use the EigCG algorithm[32], and for the latter we also employ low-mode

averaging[31].

N3s ×Nt Action as (fm) mπL mπT Lattice Generation Measurements A Measurements B

(TF-yrs) (TF-yrs) (TF-yrs)

643 × 128 W 0.11 5.0 10.0 45 120 1800

DW 0.11 5.2 10.3 700 100 1400

963 × 192 W 0.09 6.1 12.3 380 210 3100

DW 0.09 5.9 11.8 4000 390 5100

1283 × 256 W 0.06 5.4 10.9 1940 340 5000

DW 0.07 5.9 11.8 20000 1000 13000

The resources required for calculating connected diagrams for a complete set of form factorsand the lowest three moments of structure functions and generalized parton distributionsfor the nucleon at the physical pion mass are shown in Table I. Of the observables gA, thecharge radius, the magnetization radius, the magnetic moment, and 〈x〉, gA is the leastexpensive to calculate to 3 % accuracy and 〈x〉 is the most expensive. Measurements Ashows the cost of a set of calculations in which gA is computed to 3 % accuracy and theother observables are less accurate and Measurements B shows the cost of a set of calculationsin which 〈x〉, is computed to 3 % accuracy and the other observables are more accurate.Both Wilson-clover and domain wall actions have been used extensively in nucleon structurecalculations, so we show the costs using each action. Obtaining consistent results using bothactions will strengthen the credibility of lattice results, and each action has advantages forspecific calculations. The costs of the disconnected and gluonic contributions to enable theflavor-separated hadron structure are not known sufficiently reliably to include in the table.

V. HADRON SPECTROSCOPY


The calculation of the bound state spectrum of QCD encapsulates our ability to describe thestrong interactions, and the confrontation of high-precision calculations with experimental

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measurements is a vital test of the theoretical framework. The precise calculation of themasses of the lowest-lying meson and baryon states in the spectrum represents an importantmilestone in our ability to solve QCD[33]. The calculation of the excited-state spectrumprovides an opportunity to explore in detail the dynamics of QCD, and further refine ourknowledge of the theory.

The experimental investigation of the excited states of QCD has undergone a resurgence:the observation of new states in the Charmonium system at Belle and at BaBar, the searchfor the so-called missing baryon resonances of the quark model at CLAS at JLab@6GeV,and the flagship search for so-called exotic mesons at GlueX at the upgraded [email protected] do the apparent collective degrees of freedom arise that describe the spectrum arise,and can we identify them? What role do gluons play in the spectrum, and how are theymanifest? The work proposed here will facilitate those calculations both to describe theexisting experimental data, and to predict the outcomes of future experiments.

In contrast to electromagnetism, the “field-lines” between a quark and antiquark in QCD donot diffuse over large distances, but rather are confined to compact “flux tubes” connectingthem. A quark-antiquark pair connected by a flux tube is the simplest picture of a meson.A quark-antiquark pair with relative orbital angular momentum can only possess certainallowed values of JPC , and mesons having quantum numbers outside those allowed valuesare known as “exotics” and must have a richer structure. The hybrid hypothesis is that theseexotic quantum numbers arise from the addition of an excited gluon field, and thus exoticstates have attained the status of a “smoking gun” for gluonic degrees of freedom. A recentcalculation of the isovector meson spectrum including those of high spin and exotic quantumnumbers, suggests that the presence of exotics in a regime accessible to GlueX[34, 35], illus-trated in Figure 3, and furthermore the existence of both exotic and non-exotic “hybrids”.Calculations in the isoscalar sector revealed the hidden flavor-mixing angles describing theadmixtures of their light- and strange-quark components[36, 37], and suggested the presenceof isoscalar exotics at a mass comparable to their isovector cousins[37].

0.5

1.0

1.5

2.0

2.5

exotics

isoscalar

isovectorYM glueball

negative parity positive parity

FIG. 3. The spectrum of states of exotic quantum numbers for both isovectors and isoscalars, for

quark masses corresponding to those of a pion of mass 396 MeV[37]. These results suggest the

presence of many exotics in a region accessible to the future GlueX experiment.

Baryons, containing three quarks, are emblematic of the non-Abelian nature of QCD, andof SU(3). The search for so-called “missing resonances” focuses on whether the baryon

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0

500

1000

1500

2000

0

500

1000

1500

2000

FIG. 4. The excitation spectrum of hybrid mesons and baryons at three values of the light-quark

mass; for mesons we show m−mρ, while for baryons we show m−mN .

spectrum can be well described by a quark model, or whether an effective theory with fewerdegrees of freedom, such as a quark-diquark picture, provides a more faithful description. Arecent calculation of the low-lying nucleon and ∆ excited spectrum[38] exhibited a countingof levels consistent with the non-relativistic qqq constituent quark model and inconsistentwith a quark-diquark picture of baryon structure. In contrast to mesons, there are no“exotic” quantum numbers for baryons that demand a richer structure than that of threequarks with relative orbital angular momentum. None-the-less, the N and ∆ spectrumreveal the presence of “hybrid” baryons, in which, as in the case of mesons, the gluonicfield plays a vital structure role. Remarkably, the mechanism giving rise to such gluonicexcitations appears common to both mesons and baryons, as shown in Figure 4, where theexcitation energy of both “hybrid” mesons and “hybrid” baryons is shown.

The excited-state spectrum of QCD is characterized by states that are resonances unstableunder the strong interaction, and the spectrum is encapsulated within momentum-dependentphase shifts which may then be parametrised in terms of a mass and decay width. InEuclidean space QCD, shifts in the energy spectrum at finite volume can be related toinfinite-volume phase shifts at the corresponding scattering momenta[39, 40]. Recently,the energy dependence of the ρ resonance in ππ elastic scattering has been mapped inunprecedented detail[43], using methods developed to extract the momentum-dependentphase shifts in non-resonance I = 2, ππ scattering[41, 42].

Finally, there has been significant insight into understanding the electromagnetic propertiesof the excited states in QCD, in an approximation in which they are treated as quasi-stableparticles. The transition form factor from the nucleon to its lowest-lying excitation, theso-called Roper resonance[46], has been computed, complementing a vigorous experimentalprogram in this area. The electromagnetic transition form factors between charmoniumstates, including to those of exotic quantum numbers[47], have been calculated, indicatinga large partial width for the decay Γ(ηc1 → J/ψγ), and laying the groundwork for thosebetween mesons composed of light quarks relevant for the 12GeV upgrade of Jefferson Lab-oratory.

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The progress in our understanding of the spectrum of QCD has been contingent on theo-retical advances, computational advances, and on the USQCD facilities that have enabledus to efficiently exploit those advances. Notably, USQCD has adopted a program to gener-ate so-called anisotropic lattices[48, 49], with a finer temporal than spatial lattice spacing,that are designed to enable calculations of the excited-state spectrum at far less computa-tional cost than would be possible using isotropic lattices with sufficiently fine resolution. Anew method of constructing correlation functions, “distillation”[50], has enabled the largenumber of hadronic correlation functions needed to apply the variational method to be com-puted. Finally, the USQCD facilities project, including the provision of GPU-acceleratednodes, has provided a highly cost-optimized means of implementing the method, therebyfacilitating calculations that would otherwise not have been feasible.

B. Future Program and Resource Requirements:

The next five years present an exciting opportunity for lattice QCD calculations of thespectrum to predict the outcomes of future experiments, with the first experiments at the12GeV upgrade of Jefferson Laboratory expected in 2015, a proposed detector PANDA atthe FAIR facility in Germany, and an on-going program at COMPASS at CERN. A vibrantprogram of lattice calculations is vital to capitalize on these experimental investments:

• The spectrum of the low-lying meson resonances

Lattice calculations of the excited meson spectrum, albeit at unphysically large valuesof the u and d quark masses, already suggest the presence of hybrid states with massesaround 1.3 GeV above that of the ρ. Computations at quark masses corresponding tothe physical quark masses will enable the mass scale associated with hybrid mesonsto be confidently calculated, thus enabling the confrontation of the calculation ofthe spectrum within QCD, and experimental extractions of states in the region ofsensitivity of GlueX of 1.8 to 2.7 GeV.

With decreasing pion mass, and with increasing excitation in the spectrum, the statesof interest are resonances unstable under the strong interaction, and furthermore havemulti-channel decay modes for which the methods detailed in ref. [39, 40] are notapplicable, or require extensions. The development and application of methods toinclude coupled-channel that appear above the inelastic threshold[51–56] and multi-hadron final states, will be key to the implementation of this program.

The lattice calculations of the spectrum of isoscalar mesons address the question ofwhy some states are near ideally mixed, whilst others have a completely differentadmixture. Calculations at the physical quark masses will enable the mixing patternsto be discerned, notably for hybrid states, providing predictions for the decay modesthat can be tested in experiment.

These calculations will be key to capitalizing on the 2018 NSAC PerformanceMeasure HP15 to complete the first results on the search for exotic mesons usingphoton beams.

• Radiative transitions to excited mesons

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The GlueX experiment aims to photoproduce mesons and in particular those of ex-otic quantum numbers. The use of photons as a means to preferentially producehybrid states is bolstered both by phenomenological calculations, and by lattice cal-culations in charmonium indicating the radiative couplings to hybrid mesons are notsmall. Calculations of the photocouplings for light quarks will inform the expectedphotoproduction rates in experiment.

• The spectrum of excited baryon resonances

The excited-state spectrum of all the excited baryons that can be constructed fromthe u, d and s quarks will elucidate the role of quark flavor and mass in the spectrumof QCD. For the Ξ and Ω in particular, containing two and three s quarks respectively,our knowledge is very limited, with only a few states experimentally established andscant knowledge as to their properties; the production of such “very strange” baryons isproposed with both the CLAS12 and GlueX detectors at the 12 GeV upgrade of JLab.Precise calculations of the N∗ spectrum at the physical pion masses will fully capitalizeon the experimental baryon resonance program and the achievement of the 2009NSAC Performance Measure HP3. Finally, a knowledge of the decay modes andwidths of excited states can address key questions that might guide experiment, suchas whether the doubly strange Cascades are really narrower than baryons composedof the light quarks.

• Electromagnetic properties of baryon resonances

Understanding the structure of the nucleon is not limited to mapping out the charge ormomentum distributions of its constituents in its ground states; as the hydrogen atomhas shown, and understanding of the structure of a bound state and of its resonancesgo hand-in-hand. Calculations of the γNN∗ electrocouplings at Q2 < 5 GeV2 will elu-cidate the important role of pionic degrees of freedom, whilst calculations at high Q2

will explore the transition from the large-distance scales with confinement-dominateddynamics, to the short-distance scales with perturbative-dominated dynamics. Fur-thermore, a knowledge of the electromagnetic properties of excited states may provideinsight as to the quark and gluon make up of such states.

The resources needed to compute the low-lying spectrum for both mesons and baryons,together with some of their electromagnetic properties at lower momentum transfers, is out-lined in Table II. The calculations are performed on anisotropic clover lattices that have thefine temporal lattice spacing of at = 0.035 fm enabling the resolution of the many low-lyingenergy levels in the spectral function required; the longer-term approach for spectroscopywill be to use isotropic lattices with spacings at or below a = 0.035 fm.

VI. HADRONIC INTERACTIONS, NUCLEAR FORCES AND NUCLEI


The strong interactions among baryons are key to every aspect of our existence. The two- andhigher-body interactions among protons and neutrons, along with the electroweak interac-tions, produce the spectrum of nuclei and the complex chains of nuclear reactions responsible

14

TABLE II. Resources, in Tflops-years (TF-yrs), required to calculate the low-lying excited-state

spectrum for mesons and baryons, and the electromagnetic properties of the lowest-lying states,

using an anisotropic clover action with a temporal lattice spacing of at = 0.035 fm.

N3s ×Nt as (fm) at (fm) mπ (MeV) mπL mπT Trajectories Lattice Generation Measurements

(TF-yrs) (TF-yrs)

323 × 512 0.12 0.035 200 3.8 17.6 104 44 550

483 × 512 0.12 0.035 200 7.7 17.6 104 592 1305

483 × 512 0.12 0.035 140 4.0 12.3 104 411 1515

643 × 512 0.12 0.035 140 5.4 12.3 104 1210 3270

643 × 512 0.09 0.035 140 4.0 12.3 104 1210 3980

963 × 512 0.09 0.035 140 6.0 12.3 104 5531 10510

for the production of the elements forming the periodic table at the earliest times of our uni-verse, in the stellar environments that follow, and in reactors and our laboratories. Decadesof experimental effort have led to the precise measurement of the nucleon-nucleon scatteringcross sections over a wide range of energies, which have given rise to the modern nuclearforces. These experimentally determined two-body forces, encoded by modern nucleon-nucleon potentials such as the chiral potentials [57], when supplemented with three-bodyinteractions, and implemented using the renormalization group [58], provide the cornerstoneof our theoretical description of nuclei and their interactions. Coordinated efforts by the USnuclear physics community during the last several years to develop the computational tech-nology to perform convergent nuclear structure calculations show that three-nucleon forcesare required to determine the structure of light nuclei. Given the size of the three-nucleonand higher-body interactions, there is considerable uncertainty in their form, and hence inthe predictions for systems for which there is little or no experimental guidance, such asarise in extreme nuclear environments [59] that can be found, for instance, in the interior ofcore-collapse supernova. One of the motivations for building FRIB (Facility for Rare IsotopeBeams) is to better constrain the multi-neutron interactions from very neutron-rich nuclei.

As the standard model of particle physics is responsible for all nuclei and their interactions,Lattice QCD holds the promise of providing the long sought rigorous underpinnings ofnuclear physics. We anticipate that nuclear interactions will be systematically refined bylattice QCD calculations, and the uncertainties associated with future predictions of nuclearprocesses will be fully quantified. Equally important, lattice QCD allows for the quark-mass dependence of nuclear structure and the nuclear forces to be precisely determined,thereby providing a detailed exploration of fine-tunings that define our universe (in additionto being necessary for a complete determination of the chiral nuclear forces). Serious effortsto determine the properties and interactions of the light nuclei, leading to nuclear forceswhich can be propagated throughout the periodic table, have been underway for nearly adecade. Significant progress has been made toward calculating the lowest lying states ins-shell nuclei and determining the scattering parameters in the two-baryon systems. Thesecalculations have been performed with the clover action at one lattice spacing in the isospinlimit and without electromagnetism by three different lattice collaborations. The deuteronbinding energy has been determined at pion masses of mπ ∼ 400, 500 and 800 MeV [60–66]with presently insufficient precision. Similar calculations have shown the di-neutron to be

15

bound at these heavier pion masses, and all of the two-baryon octet channels contain a boundstate at the SU(3) symmetry point. The deepest bound state at the heavier pion masses isthe H-dibaryon [67–69], as conjectured in the 1970’s. The form of chiral extrapolations issufficiently uncertain so that a reliable prediction for the H-dibaryon cannot presently bemade at the physical pion mass. After significant developments in algorithms to efficientlyevaluate Wick contractions [70], correlation functions for systems 2 < A < 6 can now beconstructed, and have lead to the first calculations of 3He, 4He. Further developments havebeen possible for α-cluster nuclei. Such calculations have been extended to hypernuclei, anda first comprehensive study of s-shell nuclei and hypernuclei has recently been performed [65].Figure 5 shows the lowest-lying levels in s-shell nuclei and hypernuclei at a pion mass ofmπ ∼ 800 MeV. The binding energy of the deuteron at the heavier pion masses indicates

−200

−180

−160

−140

−120

−100

−80

−60

−40

−20

0

−B

[MeV

]

1+ 0+

1+

0+

1+

0+

12

+ 12

+

32

+

12

+

32

+

0+ 0+

0+

0+

d nn nΣ H-dib nΞ3He 3ΛH 3

ΛHe 3ΣHe4He 4

ΛHe 4ΛΛ He

s = 0 s = −1 s = −2

2-body3-body4-body

FIG. 5. The calculated low-lying spectrum of the lightest nuclei and hypernuclei at a pion mass of

mπ ∼ 800 MeV [65].

that the deuteron remains “unnatural” away from the physical pion mass as its bindingmomentum is much smaller than the inverse range of the nuclear force. In addition tointeractions of the lowest-lying octet of baryons, a calculation of the Ω−Ω− interactionshas been recently performed [71], where small scattering lengths are found. Analogouscalculations at a smaller lattice spacing, so that the lattice spacing dependence can beparametrically reduced, are planned. Further, calculations at lighter pion masses are plannedto be able to make first post-dictions of the binding of nuclei at the physical light-quarkmasses, and also to further refine the light-quark mass-dependence of nuclear observables.These post-dictions will verify the Lattice QCD technology, and the uncertainties in thepredictions for the structure of hypernuclei, and for the nuclear forces including the multi-neutron interactions, can be fully quantified.

The nucleon-nucleon scattering lengths in both spin-channels have been determined at these

16

same heavier pion masses, but the precision of the calculations has not as yet allowed forextractions of the effective range, or higher terms in the effective range expansion. Thissituation is on the verge of changing, with the first calculations of the nucleon-nucleonscattering parameters expected in the near future at the SU(3) symmetric point. Whilethe results presented by the HALQCD collaboration provide a reliable determination of thephase-shifts at the energies of the states in the lattice calculation, they are unreliable awayfrom these energies, and hence the potentials [62–64] that they extract cannot be used inmany-body calculations to provide reliable results.

-40

-35

-30

-25

-20

-15

-10

-5

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

FIG. 6. The I = 2 π+π+ elastic l = 0 (red) and l = 2 (blue) phase shifts as a function of squared

momentum at a pion mass of mπ = 396 MeV. The experimental data are shown in grey, and the

constrained analysis using the Roy equations is shown as the black line and grey band. For the

lattice calculation at this pion mass the entire region is elastic, while for the experimental data

only p2cm < 0.058 GeV is elastic [72].

The π+π+ system is the simplest with which to explore hadronic interactions due to the easeof calculability of mesonic systems compared to baryonic systems. There have been a numberof precise studies of the s-wave scattering length in this system, but recently calculationshave been extended, with advances in algorithms and in source structures, to phase shiftsbelow the inelastic threshold [41, 42, 72, 73]. Figure 6 shows the results of calculations ofthe l = 0 and l = 2 phase shifts calculated by the JLab group [72]. Extrapolation from theheavy pion masses of the calculations to the physical pion mass predict phase shifts thatare consistent with experiment and are more precise for much of the kinematic regime [42].As excited hadronic states are resonances in channels of two or more hadrons, the ability todetermine phase-shifts from the lattice energy eigenvalues is critical in the isolation of suchstates, and for theoretical predictions of the excited hadron spectrum.

The low-energy interactions between hyperons and neutrons dictates the onset of strangenessin dense matter such as that which may be created in core-collapse supernova. With suffi-ciently attractive interactions, the strange mesons and baryons become energetically favored

17

in the dense medium. Recent calculations of the nΣ− interactions at unphysical quark masseshave been extrapolated to the physical point using effective field theory. Diagonalizationof the effective field theory Hamiltonian in the finite lattice volume provided a direct com-parison with the results of Luscher’s method, and agreement was found between the twomethods within uncertainties. Weinberg’s power counting was used to perform the quarkmass extrapolation. The predicted phase-shifts agree with those determined from the avail-able experimental data. It is projected that the next generation of Lattice QCD calculationswill produce results with smaller uncertainties than those obtained from experiment, lead-ing to a quantitative improvement of theoretical predictions in these extreme astrophysicalenvironments.

Lattice QCD calculations of Bose-Einstein condensates have been extended to systems com-prised of order eighty mesons [74–77]. Analysis of these systems, when combined with theprecision two-meson calculations, have led to a determination of the three-meson interac-tion [74]. A surprising result is that tree-level χPT describes these systems even at theheavy light-quark masses, and hence tree-level χPT should provide reasonable estimates ofthe properties of a kaon-condensed phase that may form in dense matter. Recursion algo-rithms have been developed to allow for the Wick contractions contributing to a system withN mesons to be related to those for a system with N − 1 mesons, requiring only a smallnumber of operations [78].

With the exception of the threshold meson scattering parameters, all of the calculationsof nuclear interactions have been performed at a single lattice spacing, and at light-quarkmasses that produce a pion with a mass greater than mπ ∼ 250 MeV. This has been dictatedby the computational resources that have been available to the program to date. Calcula-tions involving nucleons are more complex than calculations involving mesons, such as thoserelevant to particle physics, and face different numerical challenges. While removing thesystematic uncertainty introduced by a finite lattice spacing is crucial, valuable informationabout the nuclear forces can be extracted from calculations away from the physical pionmass, such as quantifying the fine-tunings that permeate nuclear physics, and the differen-tiation of various mesonic contributions to interactions and their subsequent contributionto inelastic processes. The calculations that have been performed to date indicate thatthe two-nucleon systems remain unnatural away from the physical values of the light-quarkmasses, a somewhat surprising result.


The connection between nuclear physics and the underlying theory of fundamental forceswill be solidified during the next decade using Lattice QCD. Technology will be developedto reliably calculate the properties and interactions of light nuclei and to quantify, and sys-tematically remove, the uncertainties in such calculations. It is this rationale that underpinsthe 2014 NSAC Performance Measure HP10. Fundamental questions regarding thedependence of nuclei and their interactions on the fundamental parameters of nature, andthe fine-tunings in nuclear physics, will be answered along the way. Simultaneously, thebehavior of exotic matter, such as kaon-condensed matter or hyperonic matter, that is ex-tremely difficult, or impossible, to access in the laboratory but may play an important role

18

in extreme environments, will be quantified. It is exciting to note that the upcoming ex-perimental programs at J-PARC [96], FAIR [79], the Thomas Jefferson laboratory, and therelativistic heavy ion experiments at BNL and CERN, may be able to observe or constrainthe simplest exotic systems.

A suite of baryon-baryon, multi-baryon and multi-meson calculations must be performed atthe physical light-quark masses, and with a range of volumes and lattice spacings in order toquantify the systematic uncertainties associated with using a finite space-time grid. This willprovide the first calculations of nuclear forces from QCD that can be compared directly withthose of nature, which can then used to make reliable predictions for nuclear systems, thoughwithout electromagnetism or isospin breaking. The substantial hierarchy of energy-scalescontributing to the correlation functions that are used to extract energy-levels and S-matrixelements continues to present a challenge to the Lattice QCD calculations and requiresfurther algorithmic developments. The ongoing efforts to extend the range of calculations tomulti-nucleon systems, for instance to p-shell nuclei, requires developments in the algorithmsused to calculate Wick-contractions. Once the energy-levels of these multi-hadron systemscan be determined with precision, the matrix elements of electroweak operators betweensuch states can be calculated. Further, Lattice QCD will be used to make predictions forthe interactions between strange baryons and nucleons, and the structure of the lightesthypernuclei, that may be tested in the upcoming experimental programs.

The key research drivers in this area of research are:

• Nucleon-Nucleon Interactions and the Deuteron

The nucleon-nucleon scattering phase-shifts at low- and intermediate-energies will becalculated over a range the light-quark masses. This will provide a verification ofthe Lattice QCD technique for the nuclear forces and the associated deployed tech-nology. It will also isolate and refine the chiral nuclear potentials that are used innuclear many-body calculations, providing a rigorous underpinning to nuclear many-body calculations and allowing for the fully quantification of the uncertainties of suchcalculations. The complete decomposition of the chiral nuclear forces requires calcu-lations at unphysical light-quark masses. In addition, calculations at unphysical pionmasses will also quantify the fundamental parameter range that generates a finely-tuned nuclear physics. While the deuteron is the simplest nucleus, it is also the mostchallenging to generate at the physical light-quark masses due to its shallow binding.First post-dictions of the deuteron binding energy will be made during the next fiveyears. One of the challenging issues to be faced as the pion mass is reduced towardits physical value is the approach of the scattering lengths to very large values inboth channels, but particularly in the spin-singlet channel. This is shown explicitlyin Figure 8, along with currently allowed effective-field-theory extrapolations of theLattice QCD calculations [60]. The calculations at mπ ∼ 200 MeV will be crucial indetermining the form of the chiral extrapolation to the physical light-quark masses.

• Hyperon-Nucleon Interactions and the Core of Neutron Stars

The composition of nuclear matter at densities somewhat above that found in thecenter of nuclei is uncertain, and this uncertainty limits our present ability to predictthe evolution of explosive stellar systems, such as core-collapse supernova. Reducing

19

FIG. 7. Estimates of the resources required to accomplish calculations of importance to the nuclear

physics program. The 2018 “effective” estimate is based upon Moore’s law increases from present

resources, with a range estimated from possible algorithmic advances - the left most corresponds

to no algorithmic improvements while the right most corresponds to an extrapolation from present

day advances.

the presently significant uncertainty in the forces between hyperons and nucleons,and also amongst hyperons, will greatly improve our predictive capabilities in suchenvironments. With verification of the technology in the nucleon-nucleon sector, thepredictions for the hyperon-nucleon forces will have fully quantified uncertainties.

• S-Shell Nuclei and Hypernuclei, and their Interactions

First calculations of S-shell nuclei and hypernuclei, albeit at unphysical pion masses,have recently been performed. These nuclei will be determined at smaller lattice spac-ings and over a range of light-quark masses down to the physical pion mass. These cal-culations will enable the construction of effective interactions, including multi-nucleonforces, that can be used to calculate processes the greatly extend the range of theLattice QCD calculations themselves. While extracting the ground states have provedpossible through algorithmic advances, the excited states have significantly larger un-certainties and isolating them from excitations of continuum states in the lattice vol-ume remains a challenge. It is these continuum states that will yield the interactions ofclusters of these nuclei, and so further developments in source structures are requiredto isolate scattering states, in addition to the formal developments that are beginning

20

FIG. 8. The behavior of the nucleon-nucleon scattering length in the spin-singlet channel as a

function of the pion mass [60] (in the isospin limit and without electromagnetic interactions). W-

NLO denotes effective field theory with Weinberg’s power counting, while BBSvK denotes that

with BBSvK’s power counting. “This work” denotes the calculations of Ref. [60], which are the

only calculations within the range of validity of the effective field theory to date, and the lightest

pion mass may also be too large for convergence of the effective field theory.

to be addressed.

• P-Shell Nuclei and Hypernuclei

Lattice QCD calculations of p-shell nuclei have not been accomplished yet, limitedby contraction algorithm developments. Algorithms are currently under developmentto extend calculations into the p-shell in the near future. First calculations of suchnuclei are planned in the near future, and will be extended in parallel with those ofthe s-shell nuclei.

• α-Cluster Nuclei, 12C and the Hoyle State

An algorithm has been recently developed [70] to calculate lattice QCD correlationfunctions for α-cluster nuclei, and it has been applied to correlation functions fornuclei up to 28Si. This technology will be developed and refined to calculate suchnuclei, with a focus on 12C and the finely-tuned Hoyle state, along with the analogousstate in 16O, that is responsible for producing sufficient carbon to permit life on ourplanet. This research thrust ties in closely with recent advances in nuclear many-body calculations of 12C and the Hoyle State [80]. Presently, this is an exceptionallychallenging problem for lattice QCD, and is secondary to the precision determination

21

TABLE III. Resources, in Tflops-years (TF-yrs), required to calculate the low-lying states in nu-

clei and hypernuclei at the physical pion mass with full quantification of uncertainties using the

clover action. These estimates do not include strong isospin breaking or electromagnetism. A “∗”

indicates that the multigrid algorithm has been assumed in these resource requirements. Given the

magnitude of these resource requirements, it is anticipated that only some of these calculations will

be completed during the next five years.

N3s ×Nt b (fm) mπ (MeV) mπL mπT Trajectories Lattice Generation Measurements∗

(TF-yrs) (TF-yrs)

643 × 128 0.09 200 5.8 11.7 104 43 90

963 × 128 0.09 200 8.8 11.7 104 200 400

963 × 256 0.06 200 5.8 15.6 104 560 1100

1283 × 256 0.06 200 7.8 15.6 104 1650 3300

963 × 192 0.09 140 6.1 12.3 104 380 760

1283 × 256 0.09 140 8.2 16.4 104 1620 3200

1283 × 256 0.06 140 5.5 10.9 104 1940 3800

1923 × 384 0.06 140 8.2 16.4 104 14700 30000

of the chiral nuclear forces from the nucleon-nucleon interaction, s-shell nuclei andlighter p-shell nuclei.

Previously acquired resources have been used to generate gauge-field configurations usingthe isotropic clover action at one lattice spacing, two pion masses and in three volumes atthe SU(3) symmetric point, and to perform high-statistics calculations of nuclear correlationfunctions. The 2013 resources will be used to generate configurations and nuclear correla-tion functions at a pion mass of mπ ∼ 400 MeV. Current estimates, as shown in Figure 7,make clear that Lattice QCD calculations that directly include isospin breaking and electro-magnetism directly into the gauge field evolution cannot be accomplished without exa-scalecomputing resources. However, in nuclear physics, these effects can be approximately in-cluded post facto with peta-scale resources. During the next five years, if resources availableto this area of research increase with Moore’s Law, calculations of s-shell and p-shell nu-clei will be performed at a pion mass of mπ ∼ 200 MeV, as seen from Table III. This willallow for predictions of the periodic table of elements at this pion mass through matchingto existing and developing nuclear many-body calculations through collaboration with thebroader nuclear physics community. Further, using the effective field theory techniques thathave been developed during the last two decades, extrapolation to the physical light-quarkmasses will be performed with the uncertainties introduced in the extrapolation quantified.While the chiral nuclear forces are expected to be perturbatively close to those of nature,predictions for the nucleon-nucleon scattering lengths will likely suffer from significant un-certainties as both two-nucleon systems are near unitarity at the physical light-quark masses.When used in nuclear many-body calculations of systems that are not finely-tuned, theseforces are expected to produce results that can be extrapolated to the physical light quarkmasses with small and controlled uncertainties.

Were the available resources to scale faster than Moore’s law, and if developments in al-gorithms continue at the present rate, we expect to perform the first calculations of nuclei

22

at the physical point with two or more lattice spacings and in multiple volumes. The re-sources requirements to accomplish these objectives are shown in Table III; note that thecost of the lattice generation on the smaller lattice volume at each lattice spacing can beshared with those generated for hadron structure. These calculations will be sufficient toquantify all uncertainties associated with this technology, and the post facto inclusion ofisospin breaking and electromagnetism, will allow for direct comparisons with experiment.Once the technology is verified, predictions of processes that are difficult, or impossible, todetermine experimentally can be performed with a full quantification of uncertainties.

VII. FUNDAMENTAL SYMMETRIES


Research efforts to uncover particles and symmetries beyond those of the standard modelof the strong and electroweak interactions are multi-pronged, and key parts not only of thenuclear-physics program but of the high-energy-physics effort; some of the topics below arediscussed in further detail in the companion white paper Lattice Gauge Theory for Physicson the Intensity Frontier.

One of the approaches in this effort is to perform precision measurements of the properties ofknown particles and there is an ongoing experimental program focused on the magnetic mo-ment of the muon. The E821 experiment at Brookhaven National Laboratory has measuredthe muon g-2 with an uncertainty of 0.7 ppm, which deviates from the theoretical calculationby >∼ 3σ. The approved E-989 experiment at Fermilab is designed to reduce the uncertaintyin g-2 below 0.14 ppm, either verifying the discrepancy with theory or resolving it. A majoruncertainty in the theoretical calculation arises from strong interactions through quantumloops. Exploratory Lattice QCD calculations of the muon g−2 are underway to understandhow to calculate the strong interaction contributions[81–83], and the 2011 Kenneth Wil-son Prize [95] was awarded to Dru Renner of the European Twisted Mass Collaboration [84]for calculations of the leading contributions to the magnetic moments of the leptons.

Nature is very nearly invariant under certain symmetry transformations, and the conse-quences of the slight non-invariance can have widespread implications. A well known ex-ample is CP-violation, where the combined operation of charge-conjugation, C, and spatial-inversion, P, is known to be slightly violated, and without CP-violation, the present-daymatter and antimatter asymmetry of the universe would not exist. The nEDM collabora-tion is preparing to measure the electric dipole moment (edm) of the neutron [97], a quantitythat vanishes in the absence of time-reversal, T, violation (equivalent to CP-violation whenCPT-invariance is exact), with a precision of δdn ∼ 3 × 10−28 e cm at the Spallation Neu-tron Source (SNS) at Oak Ridge National Laboratory; the behavior is illustrated in Figure 9.There have been a number of efforts, part of the broader effort to support the 2020 NSACPerformance Measure FI15, to calculate the neutron edm arising from the QCD θ-termusing Lattice QCD [85, 86], but such calculations are found to be sensitive to the topologicalstructure of the gluon fields.

During 2010, the first Lattice QCD calculation of nuclear parity violation was performed ata pion mass of mπ ∼ 390 MeV [87], in which the “connected diagrams” contributing to the

23

FIG. 9. A cartoon showing the behavior of the neutron electric dipole moment under time-reversal.

weak one-pion-nucleon vertex, h1πNN , were determined, illustrated in Figure 10. Its valueat unphysically large light-quark masses was found to be consistent with the unnaturallysmall value extracted from theoretical analyses of a number of experimental measurements.An ongoing experimental effort by the NPDGamma Collaboration [88] at the SNS promisesto greatly reduce the experimental uncertainties in the value of h1πNN , likely in a similartime-frame to the Lattice QCD calculations.

In anticipation of precise experimental constraints on the structure of the four-Fermi op-erators contributing to the β-decay of ultra-cold neutrons, lattice QCD calculations areunderway to determine the strong interaction contributions to the matrix elements of suchoperators [89]. This is closely tied to the calculations of the structure of the nucleon, andprovides constraints on physics beyond the standard model of electroweak interactions.

Observation of a transition of neutrons to antineutrons could provide distinct evidence forbaryon number violation from beyond the Standard Model physics [90]. This hypothesizedprocess, violating the baryon number by two units, can be observed through annihilationof the resulting antineutron. Prospects for its experimental detection include large scaleproton decay and dark matter searches (in-nuclei oscillation), as well as free-neutron ex-periments [91]. For a wide range of grand unified theories (GUTs), the prediction for theoscillation period is between 109 and 1011 seconds [92]. These estimates include uncertaintiesfrom matrix elements of the effective six-quark operators between neutron and antineutron.Such matrix elements may be evaluated on a lattice, eliminating uncertainty from the oscil-lation period estimates and sensitivity of current and future experiments. Furthermore, theyare free of contributions from so-called disconnected diagrams and hence of their associateduncertatinties. Some initial calculations have been performed to test the computationalscheme [93].


The currently exploratory calculations of nuclear parity violation, will be greatly refinedwith the inclusion of quark-loop contributions. While encouraging, the present calculationsare far from conclusive even at the heavy pion mass. At lighter quark masses, with lowest-lying odd parity state unstable against pion decay, a different calculation strategy will be

24

FIG. 10. Constraints on nuclear parity-violating interactions including the recent Lattice QCD

result [87] (shaded (red) region), and the experimental 1σ uncertainty ellipse (gray).

required, likely along the lines of that described in Ref. [94]. A more complete programwill involve direct calculations of parity violation in the multi-nucleon sector, and with therecent developments in the multi-nucleon sector (outlined previously in this document), suchcalculations are currently being planned.

• Electric Dipole Moments of the Neutron, Proton and Deuteron

The neutron edm calculations will be performed at or near the physical quark masses.In addition to the contribution from the dimension-4 operator - the QCD θ-term,formally subleading contributions from dimension-5 operators will be evaluated as itmay be the case that the θ-term is dynamically forced to vanish. The calculationswill be extended to the proton and deuteron due to the experimental possibilities ofmeasuring their edms at beam facilities.

• Strong Interaction Contributions to the Muon Magnetic Moment

With the significant experimental program to more precisely measure the magneticmoment of the muon, the lattice community will be focusing on calculations to reduce

25

the uncertainty in the contribution from the strong interactions. Efforts will be madeto synchronize theoretical progress with that of the experimental program. The INThosted a initial workshop on this matter in the recent past, and close connectionsbetween the lattice community, phenomenologists and experimentalists will be fosteredgoing forward. This subject will also be a thrust in the lattice QCD at the IntensityFrontier area, as described in that Whitepaper.

Estimates of the computational resources required to accomplish the goals outlined in thissection are shown in Figure 11.

FIG. 11. Estimates of the resources required to accomplish calculations of importance to the Fun-

damental Symmetries program. The 2018 “effective” estimate is based upon Moore’s law increases

from present resources, with a range estimated from possible algorithmic advances - the left most

corresponds to no algorithmic improvements while the right most corresponds to an extrapolation

from present day advances.

VIII. SUMMARY

Lattice QCD, a technique to numerically evaluate the QCD path integral, is poised totransform the field of nuclear physics during the next five years. Not only will it provide

26

a fundamental description of the nucleon, both its structure and its excitation spectrum,and the spectrum of exotics excitations of matter, it will also provide a first principlescalculation of the spectrum and interactions of the light nuclei and hypernuclei. Withsufficient computational resources, the first calculations of many quantities of importance fornuclear physics will be performed at the physical light-quark masses. In addition to providinga direct connection between QCD and low-energy nuclear physics, the results will be used torefine and constrain hadronic-level nuclear many-body calculations, effective field theories ofmesons, baryons and multi-baryon systems, and also phenomenological descriptions of thestructure of the mesons and baryons.

IX. ACKNOWLEDGMENTS

We gratefully acknowledge contributions to this document from colleagues within theUSQCD collaboration, and in particular Jo Dudek, Robert Edwards, Balint Joo, Keh-Fei Liu, and Sergei Syritsyn. We appreciate the very insightful and informative commentsof Matthias Burkardt, Volker Crede, Ulf-G. Meissner, Witold Nazarewicz, and James Vary.

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